Consider a population with mean 𝜇 and variance 𝜎 2 < ∞ . You are comparing two estimators 𝜇 ̂ 1 and 𝜇 ̂ 2 for the mean of the population 𝜇 , with the following expected values and variances 𝐸 ( 𝜇 ̂ 1 ) = 𝜇 ; 𝑉 ( 𝜇 ̂ 1 ) = 4 ; 𝐸 ( 𝜇 ̂ 1 ) = 𝜇 + 1 ; 𝑉 ( 𝜇 ̂ 2 ) = 1 . We also know that the covariance between the two estimators is 𝐶 𝑂 𝑉 ( 𝜇 ̂ 1 , 𝜇 ̂ 2 ) = − 2 . Now consider a new estimator that combines the two previous ones 𝜇 ̂ 3 = 2 5 𝜇 ̂ 1 + 3 5 𝜇 ̂ 2 . Then the variance 𝑉 ( 𝜇 ̂ 3 ) of 𝜇 ̂ 3 is 单项选择题
A
0.04
B
2.2
C
𝜎 2 + 1
D
1
E
𝜎 2 -2
F
1.24
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Consider the likelihood of an i.i.d. sample from a Bernoulli population with parameter 𝑝 𝐿 ( 𝑥 1 , . . . , 𝑥 𝑇 ) = ∏ 𝑡 = 1 𝑇 𝑝 𝑥 𝑡 ( 1 − 𝑝 ) 1 − 𝑥 𝑡 . If you estimate the parameter 𝑝 using a Maximum Likelihood estimator, you obtain the point estimate 𝑝 ̂ = 1 𝑇 ∑ 𝑡 = 1 𝑇 𝑥 𝑡 , which corresponds to the sample mean. We know that for a Bernoulli random variable the expected value and the variance are 𝔼 ( 𝑥 𝑡 ) = 𝑝 , 𝕍 ( 𝑥 𝑡 ) = 𝑝 ( 1 − 𝑝 ) . Using this information, what is the variance of the estimator 𝕍 ( 𝑝 ̂ ) ?
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